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<p><dfn class="terminology">Example 3</dfn> (The Legendre Equation)Find two linear independent solutions for</p>
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\begin{equation*}
(1-x^2) y^{\prime \prime}-2 x y^{\prime}+\alpha (\alpha+1) y=0, \quad (\alpha &gt;-1)
\end{equation*}
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<p class="continuation">in terms of power series about <span class="process-math">\(x=0\)</span> for <span class="process-math">\(|x|&lt;1\text{.}\)</span></p>
<span class="incontext"><a href="sec5_3.html#p-220" class="internal">in-context</a></span>
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